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It is practically impossible to model a macroscopic physical system in terms of the microscopic kinematical and dynamical variables of all its particles. Thus one makes a hierarchical reduction in which this complexity is reduced to small number of collective variables. The theoretical framework for such reductions for systems is statistical mechanics or statistical physics.

One special case of hierarchical reduction is the limit of large volumes VV, in which the number of particles (of each species) per volume, N/VN/V, stays constant. This is called the thermodynamic limit in statistical physics. Under some standard assumptions like homogeneity (spacial and possibly directional) and stability (no transitory effects), there is a small number of collective variables characterizing the system. Such a description can be (and historically was) postulated as an independent self-consistent phenomenological theory even without going into the details of statistical mechanics; such a description is called equilibrium thermodynamics, which is believed to be deducible from statistical mechanics, as has been partially proved for some classes of systems. Sometimes transitional finite-time phenomena are described either statistically by studying stochastic processes or by a more elaborate hierarchical form of thermodynamics, so-called nonequilibrium thermodynamics.

One of the basic characteristics of a thermodynamical system is its temperature, which has no analogue in fundamental non-statistical physics. Other common thermodynamical variables include pressure, volume, entropy, enthalpy etc.



A formalization in terms of symplectic geometry is in chapter IV “Statistical mechanics” of

  • Jean-Marie Souriau, Structure of dynamical systems. Asymplectic view of physics . Translated from the French by C. H. Cushman-de Vries. Translation edited and with a preface by R. H. Cushman and G. M. Tuynman. Progress in Mathematics, 149. Birkhäuser Boston, Inc., Boston, MA, 1997

as well as in

Souriau model of thermodynamics has been extented for Geometric Science of Information (Koszul information geometry) with a general definition of Fisher Metric, Euler-Poincaré Equation and variational definition of Souriau Thermodynamics, as given in:

  • Frederic Barbaresco, Koszul Information Geometry and Souriau Geometric, Temperature / Capacity of Lie Group Thermodynamics, MDPI Entropy, n°16, pp. 4521-4565, August 2014. (

  • Frederic Barbaresco, Symplectic Structure of Information Geometry: Fisher Metric and Euler-Poincaré Equation of Souriau Lie Group Thermodynamics, GSI’15,Vol.9389, Springer LCNS, pp. 529-540, 2015. (

See also

For an thorough introduction to common misconceptions at an elementary level:

  • John Denker. Modern Thermodynamics. web, pdf.

A survey of irreversible thermodynamics is in

  • Ivan Vavruch, Conceptual problems of modern irreversible thermodynamics, Chem. Listy 96 (2002) (pdf)

For more on this see also rational thermodynamics.

  • Álvaro M. Alhambra, Lluis Masanes, Jonathan Oppenheim, Christopher Perry, Fluctuating work: from quantum thermodynamical identities to a second law equality, Phys. Rev. X 6, 041017 doi

Relativistic thermodynamics

Making sense of thermodynamics when taking into account special relativity and ultimately, possibly, general relativity (gravity) is notoriously subtle (even ignoring the issue of Bekenstein-Hawking entropy).

Shortly after the advent of the relativity theory, Planck, Hassenoerl, Einstein and others advanced separately a formulation of the thermodynamical laws in accordance with the special principle of relativity. This treatment was adopted unchanged including the first edition of this monograph. However it was shown by Ott and indepently by Arzelies, that the old formulation was not quite satisfactory, in particular because generalized forces were used instead of the true mechanical forces in the description of thermodynamical processes.

The papers of Ott and Arzelies gave rise to many controversial discussions in the literature and at the present there is no generally accepted description of relativistic thermodynamics.

(quote from Moller, The theory of relativity, 1952)

A standard textbook has been

  • Richard Tolman, Relativity, Thermodynamics and Cosmology, Oxford 1934, reprinted by Dover 1987

but Tolman’s approach has been called into question, see e.g.

  • Christian Fronsdal, Relativistic thermodynamics, 2014 (pdf)

See also

  • Nils Andersson, General Relativistic Thermo-Dynamics, survey talk 2014 (pdf)

  • Sean A. Hayward, Relativistic thermodynamics (arXiv:gr-qc/9803007)

  • Paul Frampton, Stephen D.H. Hsu, Thomas W. Kephart, David Reeb, What is the entropy of the universe?, Class. Quant. Grav.26:145005, 2009 (arXiv:0801.1847)

Further generalizations

Some formal generalizations of thermodynamical formalism include mixing time and temperature in formalisms with complex time-temperature like Matsubara formalism in QFT.

Mathematical analogies and generalizations include also

  • John Baez, Mike Stay, Algorithmic thermodynamics, pdf, cafe
  • M. Marcolli, Ryan Thorngren, Thermodynamical semirings, arXiv/1108.2874
  • M. Zinsmeister, Thermodynamic formalism and holomorphic dynamical systems, Amer. Math. Soc. 2000.
  • I. Itenberg, G. Mikhalkin, Geometry in the tropical limit, arXiv/1108.3111

Last revised on November 9, 2018 at 03:18:39. See the history of this page for a list of all contributions to it.